Tropical Complexity, Sidon Sets, and Dynamic Programming

Many dynamic programming algorithms for discrete 0-1 optimization problems are just special (recursively constructed) tropical (min,+) or (max,+) circuits. A problem is homogeneous if all its feasible solutions have the same number of 1s. Jerrum and Snir [JACM 29 (1982), pp. 874– 897] proved that tropical circuit complexity of homogeneous problems coincides with the monotone arithmetic circuit complexity of the corresponding polynomials. So, lower bounds on the monotone arithmetic circuit complexity of these polynomials yield lower bounds on the tropical complexity of the corresponding optimization problems. But the situation with non-homogeneous problems is entirely different: here the gap between their tropical and arithmetic complexities can be even exponential. In this paper, we improve two classical lower bounds for monotone arithmetic circuits—Schnorr’s bound and Hyafil–Valiant’s bound—and use these improvements to derive general lower bounds for the tropical circuit complexity of non-homogeneous optimization problems. In particular, we show that optimization problems, whose sets of feasible solutions are cover-free, have large tropical complexity.